scholarly journals Nonsolvability of the asymptotic Dirichlet problem for the p-Laplacian on Cartan–Hadamard manifolds

2016 ◽  
Vol 9 (2) ◽  
Author(s):  
Ilkka Holopainen
Keyword(s):  
Dirichlet Problem ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
3 Dimensional

AbstractWe construct, by modifying Borbély's example, a 3-dimensional Cartan–Hadamard manifold

scholarly journals The Dirichlet problem at infinity and complex analysis on Hadamard manifolds

1987 ◽  
Vol 106 ◽  
pp. 79-90
Author(s):  
Takashi Yasuoka
Keyword(s):  
Dirichlet Problem ◽  
Complex Analysis ◽  
Holomorphic Extension ◽  
Hadamard Manifold ◽  
Holomorphic Sectional Curvature ◽  
Geodesic Sphere ◽  
Hadamard Manifolds ◽  
Standard Sphere ◽  
Strictly Pseudoconvex Domain ◽  
Dirichlet Problem At Infinity

In this paper we shall study hyperbolicity of Hadamard manifolds.In Section 1 we shall define and solve the Dirichlet problem at infinity for Laplacian J, which gives a partial extension of the result of Anderson and Sullivan in Theorem 1 (cf.). In Section 2 we apply the solution of the Dirichlet problem at infinity to a complex analysis on a Kâhler Hadamard manifold whose metric restricted to every geodesic sphere is conformai to that of the standard sphere. It seems that the sphere at infinity of such a manifold admits a CR-structure. In fact we can define a CR-function at infinity on the sphere at infinity. We shall show in Theorem 2 that there exists a holomorphic extension from the sphere at infinity and it coincides with the solution of the Dirichlet problem at infinity, if the Dirichlet problem at infinity is solvable. So we see that such a manifold admits many bounded holomorphic functions. By the similar method we shall show in Theorem 3 that such a manifold is biholomorphic to a strictly pseudoconvex domain in Cn, if the holomorphic sectional curvature Kh(x) is less than −1/(1 + r(x)2), where r(x) is a distance function from a pole. Theorem 3 is a partial answer to a conjecture raised by Green and Wu.

scholarly journals The Dirichlet problem at infinity on Hadamard manifolds

1995 ◽  
Vol 138 ◽  
pp. 1-18 ◽  
Author(s):  
Hironori Kumura
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Beltrami Operator ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Sectional Curvatures ◽  
Geodesic Rays ◽  
Simply Connected ◽  
Dirichlet Problem At Infinity

Let M be an n-dimensional Hadamard manifold, that is, a complete simply connected C∞ Riemannian manifold with nonpositive sectional curvatures. Making use of geodesic rays, Eberlein and O’Neill [11] constructed a compactification = MS(∞) of M which gives a homeomorphism of (M, S(∞)) with the Euclidean pair (Bn, Sn-1). In this paper we shall study the asymptotic Dirichlet problem for the Laplace-Beltrami operator, which is stated as follows:

ON RICCI RANK OF CARTAN–HADAMARD MANIFOLDS

2002 ◽  
Vol 13 (06) ◽  
pp. 557-578
Author(s):  
DINCER GULER ◽  
FANGYANG ZHENG
Keyword(s):  
Lower Bound ◽  
Ricci Tensor ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
The Core ◽  
Maximum Rank

In this article, we prove that the maximum rank r of the Ricci tensor of a Cartan–Hadamard manifold Mn satisfies the inequality 2r - 1 ≥ n - s, where n is the dimension and s is the core number, which measures the flatness of Mn. Examples show that this lower bound is sharp.

scholarly journals On the Asymptotic Dirichlet Problem for the Minimal Hypersurface Equation in a Hadamard Manifold

2017 ◽  
Vol 47 (4) ◽  
pp. 485-501 ◽  
Author(s):  
Jean-Baptiste Casteras ◽  
Ilkka Holopainen ◽  
Jaime B. Ripoll
Keyword(s):  
Dirichlet Problem ◽  
Minimal Hypersurface ◽  
Hadamard Manifold

scholarly journals Solving Yosida inclusion problem in Hadamard manifold

2019 ◽  
Vol 9 (2) ◽  
pp. 357-366 ◽  
Author(s):  
Mohammad Dilshad
Keyword(s):  
Approximate Solution ◽  
Vector Field ◽  
Variational Inequalities ◽  
Hadamard Manifold ◽  
Inclusion Problem ◽  
Approximation Operator ◽  
Hadamard Manifolds ◽  
Yosida Approximation ◽  
Type Algorithm ◽  
Monotone Vector Field

Abstract We consider a Yosida inclusion problem in the setting of Hadamard manifolds. We study Korpelevich-type algorithm for computing the approximate solution of Yosida inclusion problem. The resolvent and Yosida approximation operator of a monotone vector field and their properties are used to prove that the sequence generated by the proposed algorithm converges to the solution of Yosida inclusion problem. An application to our problem and algorithm is presented to solve variational inequalities in Hadamard manifolds.

scholarly journals Explicit Iterative Method for Variational Inequalities on Hadamard Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Iterative Method ◽  
Variational Inequalities ◽  
New Method ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Monotonicity Results

An explicit iterative method for solving the variational inequalities on Hadamard manifold is suggested and analyzed using the auxiliary principle technique. The convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. Results can be viewed as refinement and improvement of previously known results.

scholarly journals NONVANISHING OF ALGEBRAIC ENTROPY FOR GEOMETRICALLY FINITE GROUPS OF ISOMETRIES OF HADAMARD MANIFOLDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 799-813
Author(s):  
ROGER C. ALPERIN ◽  
GENNADY A. NOSKOV
Keyword(s):  
Finite Groups ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Algebraic Entropy ◽  
Geometrically Finite ◽  
Geometrically Finite Group ◽  
Group Of Isometries ◽  
Groups Of Isometries ◽  
Uniform Exponential Growth ◽  
Geometrically Finite Groups

We prove that any nonelementary geometrically finite group of isometries of a pinched Hadamard manifold has nonzero algebraic entropy in the sense of M. Gromov. In other words it has uniform exponential growth.

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